On to the second part of my problem: to find the angle measure. Then, the radius is of course half that, or 9.725. Thus I get the equationįrom this it is quick to solve that d = 19.45. The two parts of the first cord are 8.5 and 8.5, and the two parts of the other are 5 and d − 5. My chord intersects the diameter of the circle, which is a chord too. If two chords intersect, the product of the segments of one chord equals the product of the segments of the other chord ( see proof). I didn't find any, but I did realize that I can use this theorem to solve my problem: I need to find the radius of the circle, AND the angle measure of the arc of the circle that makes the sliver's rounded part.Īt first, like I said, I searched around if there was some theorem or formula that would tell me what I needed directly. ![]() I have a chord of a circle, 17 mm in length in my example, and the other distance marked in the image is 5 mm. I have the height and the width of the "sliver". I wanted to make a kind of " moon-sliver shapes" in CorelDraw, to use as watermarks in my new books. It wasn't a textbook problem or a puzzle on some website, but a math problem I needed to solve for my own needs.įor a tiny while I thought I could find the answer online, but I didn't, so I'm writing it out in case someone else needs it - they should be able to find this solution by searching the Internet. Therefore, OD bisects AC.Today I had the opportunity to solve a real math problem involving a circle and a chord of known length in it. M is the midpoint of AC or that OD bisects AC. And if they have the same measure, we have just shown that So, therefore, we know that AM, AM, segment AM is going to be, I'm having trouble writing congruent is going to be congruent to segment CM, that these are going to And if the triangles are congruent then the corresponding ![]() I will write that triangle, AMO is congruent to triangle CMO by RSH. Going to be congruent to MC, but let me just write it this way. Use the Pythagorean theorem to establish that AM is And so what we could say is and let's just use RSH for now, but you could also say we can If you know two sides of a right triangle, the Pythagorean theorem would tell us that you could determine But another way to think about it, which is a little bit of common sense is using the Pythagorean theorem. We had thought about the RSH postulate where if you have a right triangle or two right triangles, you have a pair of sides are congruent, a pair and the hypothesis are congruent that means that the two With two right triangles, then it is enough. ![]() Now if you just had two triangles, that had two pairs of congruent sides that is not enough to establishĬongruency of the triangles. Perpendicular to segment AC and our assumptions in our given. How do I know they're right triangles? Well, they told us that segment OD is It's going to be equal to itself, it's going to be congruent to itself. OM is going to be congruent to OM and this is reflexivity. And then we also know that OM is going to be congruent to itself, it's a side in both of these triangles. ![]() Length AO is equal to OC because AO and OC, both radii. Radius going from O to C and another from A to O. And the way that I'm going to do this, is by establishing twoĬongruent triangles. So another way to think about it, it intersects AC at AC's midpoint. Perpendicular to this chord to chord AC, or two segment AC. So we have this circle called circle O, based on the point at its center, and we have the segment OD, and we're told that segment OD, is a radius of circle O, fair enough.
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